应用数学入门 Introduction to Applied Mathematics MAT00003C
问题 1.
Let $R=K\left[x_{1}, \ldots, x_{n}\right]$ be a polynomial ring over a field $K$ and let $I$ be a monomial ideal of $R$ generated by a finite set of monomials $\left{x^{v_{1}}, \ldots, x^{v_{q}}\right}$. As usual we use $x^{a}$ as an abbreviation for $x_{1}^{a_{1}} \cdots x_{n}^{a_{n}}$, where $a=\left(a_{1}, \ldots, a_{n}\right)$ is in $\mathbb{N}^{n}$. The three central objects of study here are the following blowup algebras: (a) the Rees algebra $$ R[I t]:=R \oplus I t \oplus \cdots \oplus I^{i} t^{i} \oplus \cdots \subset R[t], $$ where $t$ is a new variable, (b) the associated graded ring $$ \operatorname{gr}{I}(R):=R / I \oplus I / I^{2} \oplus \cdots \oplus I^{i} / I^{i+1} \oplus \cdots \simeq R[I t] \otimes{R}(R / I), $$
证明 .
with multiplication $$ \left(a+I^{i+1}\right)\left(b+I^{j+1}\right)=a b+I^{i+j+1} \quad\left(a \in I^{i}, b \in I^{j}\right), $$ and (c) the symbolic Rees algebra $$ R_{s}(I):=R+I^{(1)} t+I^{(2)} t^{2}+\cdots+I^{(t)} t^{i}+\cdots \subset R[t], $$ where $I^{(i)}$ is the ith symbolic power of $I$.
Let $R=k\left[x_{1}, \ldots, x_{n}\right]$ be a polynomial ring over a field $k$. Suppose $M=x_{1} a_{1} \ldots x_{n} a_{n}$ is a monomial in $R$. Then we define the polarization of $M$ to be the square-free monomial $$ \mathscr{P}(M)=x_{1,1} x_{1,2} \ldots x_{1, a_{1}} x_{2,1} \ldots x_{2, a_{2}} \ldots x_{n, 1} \ldots x_{n, a_{n}} $$ in the polynomial ring $S=k\left[x_{i, j} \mid 1 \leq i \leq n, 1 \leq j \leq a_{i}\right]$. If $I$ is an ideal of $R$ generated by monomials $M_{1}, \ldots, M_{q}$, then the polarization of $I$ is defined as: $$ P(I)=\left(P\left(M_{1}\right), \ldots, P\left(M_{q}\right)\right) $$
{Condition (b) is Riemann’s original formulation of integrability. 1 } (a) $\Rightarrow$ (b): Assuming $f$ is Riemann-integrable, let $$ \lambda=\int_{a}^{b} f $$
证明 .
For each $\nu=1, \ldots, n$, choose a sequence $\left(x_{\nu}^{k}\right)$ in $\left[a_{\nu-1}, a_{\nu}\right]$ such that $$ f\left(x_{\nu}^{k}\right) \rightarrow M_{\nu} \text { as } k \rightarrow \infty $$ then $$ \sum_{\nu=1}^{n} f\left(x_{\nu}^{k}\right) e_{\nu} \rightarrow \sum_{\nu=1}^{n} M_{\nu} e_{\nu}=S(\sigma) $$ so by $()$ we have (*) $$ \lambda-\epsilon \leq S(\sigma) \leq \lambda+\epsilon . $$
Similarly, $$ \lambda-\epsilon \leq s(\sigma) \leq \lambda+\epsilon \text {; } $$ thus $S(\sigma)$ and $s(\sigma)$ both belong to the interval $[\lambda-\epsilon, \lambda+\epsilon]$, therefore $$ W_{f}(\sigma)=S(\sigma)-s(\sigma) \leq 2 \epsilon . $$ This proves that $f$ is Riemann-integrable and since $$ S(\sigma) \rightarrow \int_{a}^{b} f \text { as } \mathrm{N}(\sigma) \rightarrow 0, $$ it is clear from $\left({ }^{* *}\right)$ that $$ \lambda=\int_{a}^{b} f . \diamond $$
conclusion holds for $k=1$. Let us proceed by induction on $k$. If $C$ has no interior, then by Theorem $6.2 .6$ it is included in a hyperplane $H=f^{-1}{c}$ for some non-zero linear form $f$ and $c \in \mathbb{R}$. Then $$ H={x: f(x) \geq c} \cap{x: f(x) \leq c}, $$ an intersection of two half-spaces. Also, $$ H=u+V:={u+v: v \in V} $$
证明 .
Let $V$ be a real vector space and $C$ a convex set in $V$. A real-valued function $f$ on $C$ is called convex iff $$ f(\lambda x+(1-\lambda) y) \leq \lambda f(x)+(1-\lambda) f(y) $$
and $c_{n} \rightarrow a, d_{n} \rightarrow b$; then $$ F\left(d_{n}\right)-F\left(c_{n}\right)=H\left(d_{n}\right)-H\left(c_{n}\right) $$ for all $n$, so in the limit we have $$ F(b)-0=H(b)-0 $$
证明 .
by the continuity of $F$ and In other words, $$ \int_{a}^{b} f=\int_{a}^{b} f \cdot \diamond $$
Let $F:[a, b] \rightarrow \mathbb{R}$ and $G:[a, b] \rightarrow \mathbb{R}$ be the indefinite upper integrals of $f$ and $g$, respectively: $$ F(x)=\int_{a}^{-x} f \text { and } G(x)=\int_{a}^{-x} g $$ for $a \leq x \leq b$. If $a<c<d<b$ then, as in the proof of $9.6 .3$, $$ \begin{aligned} &F(d)=F(c)+\int_{c}^{-d} f, \ &G(d)=G(c)+\int^{-d} g \end{aligned} $$
Proof. Let $h_{n} \rightarrow 0, h_{n} \neq 0$. Since $E$ is continuous, $E\left(h_{n}\right) \rightarrow E(0)=$ 1. Let $x_{n}=E\left(h_{n}\right)$; then $x_{n}>0, x_{n} \rightarrow 1$ and $x_{n} \neq 1$ (because $\left.h_{n} \neq 0\right)$, so by $9.5 .10$ $$ \frac{L\left(x_{n}\right)}{x_{n}-1} \rightarrow 1 $$
证明 .
That is, $$ \frac{h_{n}}{E\left(h_{n}\right)-1} \rightarrow 1 $$ whence the lemmn.
(i) Show that $A$ is an odd function: $A(-x)=-A(x)$ for all $x \in \mathbb{R}$. {Hint: $f$ is an even function.} (ii) $A$ is strictly increasing. {Hint: $\left.A^{\prime}=f .\right}$ (iii) For every positive integer $k$, $$ \frac{1}{1+k^{2}} \leq A(k)-A(k-1) \leq \frac{1}{1+(k-1)^{2}} . $$ {Hint: Integrate $f$ over the interval (iv) Let $$ s_{n}=\sum_{k=1}^{n} \frac{1}{1+k^{2}} . $$
Various ideas from real analysis can be generalized from the real line to broader or more abstract contexts. These generalizations link real analysis to other disciplines and subdisciplines. For instance, generalization of ideas like continuous functions and compactness from real analysis to metric spaces and topological spaces connects real analysis to the field of general topology, while generalization of finite-dimensional Euclidean spaces to infinite-dimensional analogs led to the concepts of Banach spaces and Hilbert spaces and, more generally to functional analysis.
实分析课后作业代写
Using the logarithmic function defined in Section $7.1$, the reciprocal of any linear polynomial can be integrated. Indeed, up to a constant multiple, such a function is given by $1 /(x-\alpha)$, where $\alpha \in \mathbb{R}$, and we have $$ \frac{d}{d x}(\ln (x-\alpha))=\frac{1}{x-\alpha} \quad \text { for } x \in \mathbb{R}, x>\alpha $$ The next question that naturally arises is whether we can integrate the reciprocal of a quadratic polynomial, say $x^{2}+a x+b$, where $a, b \in \mathbb{R}$. If this quadratic happens to be the square of a linear polynomial, say $(x-\alpha)^{2}$, then the answer is easy because $$ \frac{d}{d x}\left(\frac{-1}{x-\alpha}\right)=\frac{1}{(x-\alpha)^{2}} \quad \text { for } x \in \mathbb{R}, x \neq \alpha . $$ Further, if the quadratic factors into distinct linear factors, that is, if $x^{2}+a x+b=(x-\alpha)(x-\beta) \quad$ for some $\alpha, \beta \in \mathbb{R}, \alpha>\beta$
我们可以处理的内容但不限于实数的构造（Construction of the real numbers），实数的序关系（Order properties of the real numbers），实数的拓扑性质（Topological properties of the real numbers），数列（Sequences）、极限和收敛性（Limits and convergence）、函数的一致收敛和逐点收敛（Uniform and pointwise convergence for sequences of functions）、紧性（Compactness）、连续性（Continuity）、可微性（Differentiation）、黎曼积分（Riemann integration）、勒贝格积分（Lebesgue integration）等等。
